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Base field \(\Q(\sqrt{13}) \)

Copy content comment:Define the base number field
 
Copy content sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, -1, 1]))
 
Copy content pari:K = nfinit(Polrev(%s));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R!%s);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(Qx(%s))
 

Generator \(a\), with minimal polynomial \( x^{2} - x - 3 \); class number \(1\).

Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([K([0,0]),K([0,1]),K([1,0]),K([-23460,-7820]),K([790739,1054319])]) E.isogeny_class()
 

Rank

Copy content comment:Compute the Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content magma:Rank(E);
 

The elliptic curves in class 1089.2-a have rank \( 1 \).

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 

\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_class().graph().plot(edge_labels=True)
 

Elliptic curves in class 1089.2-a over \(\Q(\sqrt{13}) \)

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 

Isogeny class 1089.2-a contains 3 curves linked by isogenies of degrees dividing 25.

Curve label Weierstrass Coefficients
1089.2-a1 \( \bigl[0\) , \( a\) , \( 1\) , \( -7820 a - 23460\) , \( 1054319 a + 790739\bigr] \)
1089.2-a2 \( \bigl[0\) , \( a\) , \( 1\) , \( -10 a - 30\) , \( 79 a + 59\bigr] \)
1089.2-a3 \( \bigl[0\) , \( a\) , \( 1\) , \( 0\) , \( -a - 1\bigr] \)